Less than or equal set to a given matrix

Question

Suppose that $M$ is a given positive definite $n\times n$ real matrix. I'm interested in the following sets and their properties \begin{align*} S_1&\equiv\{N\text{ positive semidefinite, dimension }n\times n: M-N\text{ is positive semidefinite}\}\subset \mathbb{M}(n),\\ S_2&\equiv\{x\text{ dimension }n\times1: M-xx'\text{ is positive semidefinite}\}\subset\mathbb{R}^n. \end{align*} The problem collapses to triviality when $n=1$ but harder for me to grasp when $n\geq 2$. I will appreciate references to books and articles, with a preference for the former.